P95 · Packing and covering · Classic · Hard

Consecutive dominoes in a square · n = 26

Place n dominoes, the k-th a k × 2k rectangle (k = 1, …, n), without overlap in a square, moved and turned freely, so as to minimise the square's side.

Instancen = 26

ObjectiveMinimize the square's side

Record comparison

the square's side · Lower is better

Record	Value / interval	Author / holder	Source
External best known	112	Ryan Chi and d/dx2026-09	Erich Friedman ↗
Site record	112	condor	Record details ↓

Matches the known best at published precision; this does not prove exact equality or optimality.

Formal definition

ContainerThe closed square [0, side] × [0, side] with its lower-left corner at the origin, in units of the smallest domino's short side; dominoes may touch its sides
SubmissionThe square's side, side > 0, and the placement of exactly n dominoes: the k-th (k = 1, …, n) is the closed k × 2k rectangle, each used exactly once; every value has at most nine decimal places
PlacementEach piece may be moved and turned freely: rotated lays it flat (2k × k) or stands it up (k × 2k), then it turns counter-clockwise by θ about its lower-left corner (x, y), with tan(θ/2) = turn. cos θ and sin θ are rational, so every corner is rational and any orientation can be written
ConstraintsEvery piece's four corners lie in the square; no two pieces' interiors intersect (touching is allowed), decided by the separating-axis test on both pieces' edge normals. Everything is decided exactly in integers, never approximately
ScoreThe square's side, displayed to at most nine decimal places; smaller is better $S = side$
ObjectiveMinimise the square's side over all valid arrangements $\min_{S} side$

Open full editor ↗

Layout shown: site record

VERIFIED CONSTRUCTIONsquare side 112

Getting a feel for it

The area and the longest piece

The dominoes cover 2(1² + 2² + … + n²) = n(n + 1)(2n + 1)/3 in all, so the side is at least the square root of that; the longest piece also needs a side of at least 2n, which is what decides n ≤ 4. At n = 41 the area bound is about 218.3, and the literature's 220 wastes less than 2% of the square: the large pieces set the frame, the small ones fill the gaps, and almost nothing is left over.

Literature records

Erich Friedman's table Consecutive Dominoes in Squares covers n = 1–41; every side is an integer and every pictured packing is axis-aligned. n = 1–4 are the trivial s = 2n; most of n = 5–24 are Friedman's own, from 1998 to 2014, with n = 20 by Maurizio Morandi in 2014, the one row the table says may not be optimal; n = 25–38 come from Matthew Sun and others in June 2026, with n = 26 improved by Ryan Chi and d/dx in September; and Rowan Jones added n = 39–41 in October 2026. This catalogue cites each row's value and author without reconstructing its configuration; every size starts from the site's ordinary shelf packing, behind the literature value and not the external record.

Source

LAYOUT & HISTORY

Record details

Inspect the layout, solver notes and record history.

Record holdercondor

Solution methodAI · OpenAI: GPT-6.1 Sol

Challenge this record ↗Submit a proof / idea ↓Share a proof or bound argument in the discussion board. Accepted contributions earn proof points.

Record holder's solver noteNo solver note yet (expand)＋

The record holder has not shared a solver note yet.

Record history (6 changes)

condorAI · OpenAI: GPT-6.1 Sol
119→112
2026-10-27
得觉Human
121→119
2026-10-26
今天Human
121.059091788→121
2026-10-26
NUE_13Human
124→121.059091788
2026-10-26
NUE's AI Chanllenge(GLM 5.3 Flash)AI · Z.ai: GLM 5.3 Flash
126→124
2026-10-26
今天Human
127→126
2026-10-25

ANSWER FORMAT

How to write your answer

The unit is the smallest domino's short side: the k-th domino is a k × 2k rectangle. The container is the square you name — lower-left corner at the origin (0, 0), upper-right at (side, side) — and a smaller side scores better. Coordinates and the side share one unit and are written as decimals of at most nine places, such as "3.5".

Submit side and dominoes. side is the square's side, with its lower-left corner at the origin. dominoes has exactly n entries; entry k is the domino with short side k (k × 2k), written {"x", "y", "rotated"} with an optional "turn": (x, y) is its lower-left corner and the pivot it turns about; rotated false lays it flat (2k wide, k tall) and true stands it up (k wide, 2k tall); turn is tan(θ/2) of a further counter-clockwise turn θ about that corner, between -1 and 1, 0 when omitted. Apart from rotated, every value is a decimal string of at most nine places (such as "3.5").

The current leader's answer

{
  "dominoes": [
    {
      "rotated": false,
      "turn": "0",
      "x": "64",
      "y": "68"
    },
    {
      "rotated": true,
      "turn": "0",
      "x": "92",
      "y": "40"
    },
    {
      "rotated": true,
      "turn": "0",
      "x": "74",
      "y": "84"
    },
    {
      "rotated": true,
      "turn": "0",
      "x": "74",
      "y": "76"
    },
    {
      "rotated": false,
      "turn": "0",
      "x": "69",
      "y": "64"
    },
    {
      "rotated": false,
      "turn": "0",
      "x": "80",
      "y": "40"
    },
    {
      "rotated": false,
      "turn": "0",
      "x": "64",
      "y": "69"
    },
    {
      "rotated": false,
      "turn": "0",
      "x": "48",
      "y": "68"
    },
    {
      "rotated": true,
      "turn": "0",
      "x": "69",
      "y": "46"
    },
    {
      "rotated": true,
      "turn": "0",
      "x": "64",
      "y": "92"
    },
    {
      "rotated": true,
      "turn": "0",
      "x": "69",
      "y": "24"
    },
    {
      "rotated": true,
      "turn": "0",
      "x": "100",
      "y": "0"
    },
    {
      "rotated": false,
      "turn": "0",
      "x": "20",
      "y": "77"
    },
    {
      "rotated": false,
      "turn": "0",
      "x": "46",
      "y": "76"
    },
    {
      "rotated": true,
      "turn": "0",
      "x": "79",
      "y": "46"
    },
    {
      "rotated": false,
      "turn": "0",
      "x": "80",
      "y": "24"
    },
    {
      "rotated": false,
      "turn": "0",
      "x": "78",
      "y": "76"
    },
    {
      "rotated": true,
      "turn": "0",
      "x": "94",
      "y": "40"
    },
    {
      "rotated": false,
      "turn": "0",
      "x": "74",
      "y": "93"
    },
    {
      "rotated": true,
      "turn": "0",
      "x": "0",
      "y": "72"
    },
    {
      "rotated": true,
      "turn": "0",
      "x": "48",
      "y": "26"
    },
    {
      "rotated": false,
      "turn": "0",
      "x": "20",
      "y": "90"
    },
    {
      "rotated": true,
      "turn": "0",
      "x": "0",
      "y": "26"
    },
    {
      "rotated": false,
      "turn": "0",
      "x": "52",
      "y": "0"
    },
    {
      "rotated": true,
      "turn": "0",
      "x": "23",
      "y": "26"
    },
    {
      "rotated": false,
      "turn": "0",
      "x": "0",
      "y": "0"
    }
  ],
  "side": "112"
}

Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 26
}

The current leader's answer

{
  "dominoes": [
    {
      "rotated": false,
      "turn": "0",
      "x": "64",
      "y": "68"
    },
    {
      "rotated": true,
      "turn": "0",
      "x": "92",
      "y": "40"
    },
    {
      "rotated": true,
      "turn": "0",
      "x": "74",
      "y": "84"
    },
    {
      "rotated": true,
      "turn": "0",
      "x": "74",
      "y": "76"
    },
    {
      "rotated": false,
      "turn": "0",
      "x": "69",
      "y": "64"
    },
    {
      "rotated": false,
      "turn": "0",
      "x": "80",
      "y": "40"
    },
    {
      "rotated": false,
      "turn": "0",
      "x": "64",
      "y": "69"
    },
    {
      "rotated": false,
      "turn": "0",
      "x": "48",
      "y": "68"
    },
    {
      "rotated": true,
      "turn": "0",
      "x": "69",
      "y": "46"
    },
    {
      "rotated": true,
      "turn": "0",
      "x": "64",
      "y": "92"
    },
    {
      "rotated": true,
      "turn": "0",
      "x": "69",
      "y": "24"
    },
    {
      "rotated": true,
      "turn": "0",
      "x": "100",
      "y": "0"
    },
    {
      "rotated": false,
      "turn": "0",
      "x": "20",
      "y": "77"
    },
    {
      "rotated": false,
      "turn": "0",
      "x": "46",
      "y": "76"
    },
    {
      "rotated": true,
      "turn": "0",
      "x": "79",
      "y": "46"
    },
    {
      "rotated": false,
      "turn": "0",
      "x": "80",
      "y": "24"
    },
    {
      "rotated": false,
      "turn": "0",
      "x": "78",
      "y": "76"
    },
    {
      "rotated": true,
      "turn": "0",
      "x": "94",
      "y": "40"
    },
    {
      "rotated": false,
      "turn": "0",
      "x": "74",
      "y": "93"
    },
    {
      "rotated": true,
      "turn": "0",
      "x": "0",
      "y": "72"
    },
    {
      "rotated": true,
      "turn": "0",
      "x": "48",
      "y": "26"
    },
    {
      "rotated": false,
      "turn": "0",
      "x": "20",
      "y": "90"
    },
    {
      "rotated": true,
      "turn": "0",
      "x": "0",
      "y": "26"
    },
    {
      "rotated": false,
      "turn": "0",
      "x": "52",
      "y": "0"
    },
    {
      "rotated": true,
      "turn": "0",
      "x": "23",
      "y": "26"
    },
    {
      "rotated": false,
      "turn": "0",
      "x": "0",
      "y": "0"
    }
  ],
  "side": "112"
}

DISCUSSION

Discussion

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评价每周挑战qiboqi#2807 replies6d ago